Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues

Chen, Zhao; Grimm, Matthew; McMichael, Paul; Johnson, Charles R.

doi:10.1016/j.laa.2014.02.022

Cited by 10 publications

(33 citation statements)

References 7 publications

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“…We note that the second case of the proof importantly uses the theory of DM matrices described in [4]. The proof of the first case cannot be adapted to the second case.…”

Section: Resultsmentioning

confidence: 99%

Multiplicity lists for symmetric matrices whose graphs have few missing edges

Johnson

Zhang

2018

Linear Algebra and its Applications

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We characterize the possible lists of multiplicities occurring among the eigenvalues of real symmetric (or Hermitian) matrices whose graph is one of K n , K n less an edge, or both possibilities for K n less two edges. The lists are quite different from those for trees. Some construction techniques are developed here and additional results with more missing edges are given, including the case of several independent edges.

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“…We note that the second case of the proof importantly uses the theory of DM matrices described in [4]. The proof of the first case cannot be adapted to the second case.…”

Section: Resultsmentioning

confidence: 99%

Multiplicity lists for symmetric matrices whose graphs have few missing edges

Johnson

Zhang

2018

Linear Algebra and its Applications

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“…It turns out that cloning a vertex of a graph G with MB(G) = k results in a graph G with MB(G ) ≤ k. The following proposition is proved in Theorem 6.3 of [13], it is also implied by Corollary 4 of [2]. In [13], this is used to characterize graphs G with MB(G) = by constructing minimal such graphs (these are K , K ∪ K , K , , K , ,..., and K , ,..., , ) and constructing all the other such graphs by cloning vertices in the minimal graphs.…”

Section: Lemma 33 For Any Graph G With No Isolated Vertices and Formentioning

confidence: 91%

Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]

et al. 2020

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“…is the closed neighbourhood of v (that is, a neighbourhood of v containing v). It turns out that cloning a vertex of a graph G with M B(G) = k results in a graph G ′ with M B(G ′ ) ≤ k. The following proposition is proved in Theorem 6.3 of [12], it is also implied by Corollary 4 of [1]. In [12], this is used to characterize graphs G with M B(G) = 2 by constructing minimal such graphs (these are K 1 , K 1 ∪ K 1 , K 2,1 , K 2,2,...,2 and K 2,2,...,2,1 ) and constructing all the other such graphs by cloning vertices in the minimal graphs.…”

Section: Complete Multipartite Graphsmentioning

confidence: 91%

“…The first statement is trivial. The second statement has appeared in [7,12,13,15]. The third statement is known (for example, there is a proof in [13]), but we include a proof for completeness.…”

Section: Graphs With Two Distinct Eigenvaluesmentioning

confidence: 99%

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

et al. 2019

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Associated to a graph G is a set S(G) of all real-valued symmetric matrices whose off-diagonal entries are non-zero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in S(G) partition n; this is called a multiplicity partition.We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in S(G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in S(G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in S(G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

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Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues

Cited by 10 publications

References 7 publications

Multiplicity lists for symmetric matrices whose graphs have few missing edges

Multiplicity lists for symmetric matrices whose graphs have few missing edges

Corrigendum to “Achievable Multiplicity partitions in the Inverse Eigenvalue Problem of a graph” [Spec. Matrices 2019; 7:276-290.]

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

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